On the maximal number of du Val singularities for a K3 surface

A complex K3 surface or an algebraic K3 surface in characteristics distinct from $2$ cannot have more than $16$ disjoint nodal curves.


Convention
In this note is a compact complex surface, and in § 4 it is a K3 surface. I assume throughout that 2 ( , ℤ) has no torsion and I consider it as a lattice equipped with the integral pairing induced by the cup product. In that situation the Néron-Severi group ( ) is to be considered as a sublattice spanned by the classes of the divisors. This is a primitive sublattice of 2 ( , ℤ).
If is an integral lattice and ∈ ℚ, I denote by ( ) the same ℤ-module as but with form multiplied by .

Codes and lattices
A binary code is a linear subspace of some finite dimensional space over the field 2 which I shall identify with 2 , = dim . A vector of a code is also called a word. The number of non-zero coordinates of is its weight, ( ). The dot-product ⋅ = ∑ ∈ 2 of two vectors = ( 1 , … , ) and = ( 1 , … , ) in defines a non-degenerate symmetric bilinear form and a code is called isotropic if ⊂ ⟂ and self-dual if = ⟂ . Via the reduction modulo 2 map ∶ ℤ → 2 a code lifts to a submodule −1 of ℤ with 2 ⋅ ℤ ⊂ −1 ⊂ ℤ . The first inclusion shows that −1 is of finite index in ℤ . It inherits the structure of a lattice from ℤ equipped with its dot-product. However, it turns out to be more convenient to use a different lattice structure, namely ℤ equipped with the standard euclidean form scaled by 1 2 , which is ℤ 1 2 by the convention I adopted. This leads to the lattice (1) One easily sees that the new product on Γ has integral values if and only if is isotropic. See e. g. [Ebe94, § 1.3]. I shall make use of the so-called Reed-Muller codes whose definition runs as follows. Let be a finite set of size . Then the functions → 2 form the 2vector space 2 . Suppose that itself consists of the points of an 2 -vector space of dimension so that = 2 . I order these points as follows. Let { 0 , … , −1 } be a basis for and identify a point = with the binary expansion = ∑ −1 =0 2 of an integer between 0 and 2 − 1. The natural order gives an ordering of the points of . A function on determines a vector ( 0 , … , −1 ) ∈ 2 as follows. First note that a point ∈ determines the unique integer = ∈ [0, … , − 1] and then I set = ( ). The polynomial functions of degree on together with the zero function define a subspace of 2 and this is also the case for polynomial functions of degree ≤ . The latter define the -th order Reed-Muller code ≤ ( ) ⊂ 2 , = 2 . For an extensive treatment of these codes I refer to [Lin92,Chap. 4.5.].
Example 2.1. Take = 4 and = 1. Then = 2 4 = 16 and ≤1 ( 4 2 ) ⊂ 16 2 is generated by the 4 code words given as the rows of the following 4 × 16 matrix together with the vector with all coordinates equal to 1 arising from the constant function 1. The columns correspond to the binary expansions of the numbers 0, ..., 15 and the rows correspond to the coordinate functions 0 , 1 , 2 , 3 : (2) The code is a code of dimension +1 given by the affine linear functions on . One can view it as generated by the code ∶= 1 , consisting of the linear functions on , together with the constant function 1. This last word has weight 2 while the nonzero weights of are all 2 −1 since the characteristic function of a hyperplane interpreted as a code word has this weight. So the weights of itself (lowering the index by one) are 0, 2 −2 and 2 −1 . The code can be characterized as follows: I need a further property of the code : There is no code in 2 with the property that all coordinate subspaces of 2 of dimension 2 −1 give a code isomorphic to .
Proof. Set = 2 −1 . Visualize the code ⊂ 2 as generated by (1, … , 1) together with the rows of the ( − 1) × matrix, say , whose columns are the binary expansions of the numbers 0, … , − 1 ordered as in (2) where = 5, = 16. Suppose that I have a code with the stated properties. Then it contains at least ( − 1) vectors whose first coordinates form the matrix . I now consider the corresponding vectors in 2 forming a matrix . It suffices to consider the case = + 1 since the effect of deleting − − 1 columns from and a further column from is the same for all ≥ + 1. Suppose that the last column of corresponds to the decimal expansion of ∈ [0, … , − 1]. Then this column is the same as the -th column of . Form now the ( − 1) × matrix by deleting from a column of the submatrix with ≠ . Since all columns of are different, there is a row such that the entry of is distinct from the entry of . This implies that the -th row of the new matrix has weight 1 2 ± 1 which is a contradiction. Hence there is no such code.

Sets of nodal curves on a surface
Let be a compact complex surface containing a finite set E = { 1 , … , } of disjoint smooth rational curves with 2 = −2, = 1, … , . Such curves are also called nodal curves, since these arise as minimal resolutions of ordinary double points. Let = ℤ 1 ⦹ ⋯ ⦹ ℤ be the abstract lattice with basis the nodal classes 1 which can be identified with ℤ (−2). Its dual is given by * = , setting the lattice E (−1) is precisely the inverse image of the code E under the mod 2 map. In other words, Using that E is primitive in the lattice ( ), I arrive at an equivalent description of the code E , namely Here primitivity is used to establish the rightmost inclusion. The first consequence of this description is a bound for dim E . Since ( ) is primitive in 2 ( , ℤ), the quotient ( )∕2 ( ) injects into 2 ( , ℤ)∕2 2 ( , ℤ) = 2 ( , 2 ), a symplectic inner product space in which the image of is totally isotropic and so has dimension ≤ 1 2 2 ( ). It follows that Secondly, using the notion of an "even set of nodal curves", which, I recall, means that the sum of the nodal curves is divisible by 2 in ( ), there is a further consequence:

Applying coding theory to nodal K3 surfaces
There are severe restrictions on even sets of disjoint nodal curves on a complex K3 surface:   … , 1). Proof. The first assertion is a direct translation of Lemma 4.1 using Lemma 3.1.
The assertion for = 8 is clear.
Since the code 5 contains the word (1, … , 1), the sum of all the nodal curves is even and so any set of 16 disjoint nodal curves on a K3 surface is an even set. This reproves a result by V. V. Nikulin [Nik75]. Forming the double cover gives a complex two-torus blown up in 16 points and the quotient by the standard 2 Indeed, the canonical bundle is trivialized on by a non-zero holomorphic two-form. On the double cover it lifts as a holomorphic two-form which is non-zero outside the branch locus and descends a holomorphic two-form on , nowhere zero except maybe in the points . But a section of a line bundle can at most have zeros along a divisor and so trivializes .
involution is a Kummer surface whose minimal resolution of singularities is . Hence: Proposition 4.4. Let be a complex K3 surface containing a set E of 16 disjoint nodal curves. Then E is an even set, and is the minimal resolution of a Kummer surface. Moreover, the primitive sublattice of ( ) spanned by E is isometric to the lattice Γ 5 (−1).
The lattice spanned by 16 disjoint nodal curves on a (desingularised) Kummer surface is also called the Kummer lattice. Its characterization as the abstract lattice Γ 5 (−1) makes it possible to show the main result of this note: where is the minimal resolution of singularities of̄ . The number ( ) gives the number of disjoint nodal curves on coming from desingularizinḡ and so one finds: Corollary 4.6. If is a K3 surface which is the minimal resolution of singularities of a surface having at most du Val singularities, then ( ) ≤ 16.
In particular,̄ can have at most 16 ordinary nodes, or 1 -singularities. More generally, ( ) = 16 if there are only 1 or 2 singularities, but otherwise it is strictly smaller. For instance, one can have at most four singularities which all are of types 16 , 6 , 7 , 7 or 8 . Also note the discrepancy with the Milnor number ( ) = ∑ ( + + ). Indeed the more singularities with high Milnor number, the closer ( )∕ ( ) gets to 1 2 .
Remark 4.7. One can say more for specific types of projective K3 surfaces. For instance, a degree 6 K3 surface in ℙ 5 , necessarily a complete intersection of a quadric and a cubic, can have no more than 15 double points. See [Kal86].